Paper X — Horizons, Saturation, and Constraints on Quantum Gravity

DOI: 10.5281/zenodo.17925756 Horizons, Saturation, and Constraints on Quantum Gravity Paper X of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 1.1 Scope and limitations 2 2 Operational Singularities 2 3 Minimum Operational Resolution 2 4 Horizons as Information Boundaries 3 5 Black Hole and Cosmological Horizons 3 6 Singularities as Descriptive Breakdowns 3 7 Relation to the Reconstruction Program 4 8 Conclusion 4 DOI: 10.5281/zenodo.17925756 he Ordered-Dynamics Reconstruction Program derives the kinematical, dynamical, interactional, and measurement structure of quantum theory from operational principles without postulating spacetime geometry, collapse, or classical limits. This final paper identifies the ultimate consistency constraints imposed by bounded information density and locality. We show that unbounded spatial resolution, infinite curvature, and classical singularities correspond to regimes in which operational descriptions cease to be predictive due to unbounded informational requirements. Horizons arise as boundaries of information accessibility, finite spatial resolution emerges as an operational limit rather than a geometric postulate, and singularities are excluded as physical endpoints and reinterpreted as descriptive breakdowns. 1 Introduction The preceding papers of the Ordered-Dynamics Reconstruction Program establish a complete reconstruction of quantum kinematics, interaction structure, gauge redundancy, records, probability, and irreversibility from minimal operational assumptions. At no point have spacetime geometry, gravitational dynamics, or measurement postulates been introduced as primitives. Nevertheless, standard physical theories encounter regimes where classical descriptions predict divergences: infinite curvature, arbitrarily fine spatial resolution, and singular endpoints of evolution. Traditionally, these features are treated as indicators of new dynamics or new degrees of freedom. In this paper we show that such regimes are already constrained—and in some cases excluded—by the same operational principles underlying the earlier reconstruction. The argument does not depend on specific gravitational field equations or quantization schemes. Instead, it relies only on bounded information density, locality of influence, and the necessity of stable records. – 1 – 1.1 Scope and limitations This paper does not: •derive Einstein’s equations, •propose a specific theory of quantum gravity, •quantize spacetime geometry, •dynamically resolve classical singularities. It identifies the boundaries beyond which operationally meaningful descriptions fail and shows that these boundaries coincide with familiar horizon and singularity structures. 2 Operational Singularities Definition 1 (Operational singularity).An operational singularity is a regime in which maintaining predictive consistency would require unbounded information density, resolution, or control by a bounded observer. Remark 1.Operational singularities are defined independently of coordinate choices or curvature invariants. They concern what can be operationally described, not what formally diverges. Operational singularities are incompatible with bounded information density. Proof. Bounded information density limits the number of distinguishable states or records that can be supported in any finite region. Any regime requiring arbitrarily fine distinctions or infinite control violates this constraint and cannot be operationally maintained. 3 Minimum Operational Resolution Finite information density and locality impose a lower bound on operationally distinguishable spatial features for bounded observers. Proof. Resolving smaller spatial features requires encoding finer distinctions in records and correlations. With finite information density, such distinctions eventually exceed the available capacity. [Minimum operational resolution] For any bounded observer subject to finite information density and local operations, there exists a nonzero lower bound on spatial distinctions that can be operationally resolved. Proof. Combining finite information density with locality restricts the number of independent degrees of freedom accessible within any finite region. Below a certain scale, distinctions cannot be reliably encoded or retrieved by local operations. Remark 2.This bound is operational rather than geometric. It does not assert a universal minimum length scale, but a limit on resolvable distinctions for bounded observers. Different observers or regimes may admit different effective bounds. – 2 – 4 Horizons as Information Boundaries Definition 2 (Operational horizon).A horizon is a boundary separating regions of operational accessibility for a bounded observer, beyond which information cannot be retrieved or used to form records within finite operations. Operational horizons arise whenever influence propagation is finite and information capacity is bounded. Proof. Finite influence propagation prevents instantaneous coordination across large regions. When combined with bounded record capacity, this produces regions whose information cannot be accessed or reconstructed by any bounded observer. Remark 3.Operational horizons may coincide with geometric horizons in specific dynamical theories, but their existence here does not depend on spacetime metric structure. 5 Black Hole and Cosmological Horizons Black hole horizons correspond to operational horizons associated with loss of record accessibility due to extreme information concentration. Proof. When information becomes concentrated beyond recoverable limits, records originating within the region cannot be accessed by external observers within finite operations. This loss of accessibility defines an operational horizon. Cosmological horizons arise from finite influence propagation and bounded record formation over large-scale backgrounds. Proof. Finite propagation speed and expanding influence domains prevent complete coordination or reconstruction of distant records, yielding observer-dependent horizons. Remark 4.Both black hole and cosmological horizons arise here as manifestations of the same operational principle: loss of record accessibility under bounded influence. 6 Singularities as Descriptive Breakdowns [Singularities as descriptive breakdowns] Classical singularities correspond to limits in which operational descriptions cease to be predictive due to unbounded informational requirements. They cannot represent physical endpoints of evolution within a bounded-information theory. Proof. Approaching a classical singularity requires resolving arbitrarily fine distinctions and controlling diverging correlations. Such requirements exceed the informational capacity available to bounded observers, rendering the description incomplete. Remark 5.This result does not prohibit extreme curvature or divergent regimes. It states that such limits cannot be treated as complete physical descriptions and must be replaced by horizons, information saturation, or loss of operational access. Any consistent completion of the theory must replace singularities with informationsaturating regions, horizons, or effective bounces. – 3 – 7 Relation to the Reconstruction Program •Papers I–III establish kinematics and transport. •Paper IV constrains interaction locality. •Papers VII–VIII derive fields and gauge redundancy. •Paper IX explains records, probability, and irreversibility. •Paper X identifies the ultimate operational limits. This paper closes the reconstruction by identifying the boundaries beyond which no operationally meaningful refinement is possible. 8 Conclusion Horizons, finite operational resolution, and the exclusion of singular endpoints are not optional features added to physical theory. They arise inevitably from bounded information density, locality, and the necessity of stable records. With this result, the Ordered-Dynamics Reconstruction Program forms a closed, selfconsistent reconstruction of fundamental physics grounded entirely in operational principles. Acknowledgments The author acknowledges that this work concludes the effective field reconstruction (Series I). The derivation of the emergent substrate itself is developed in the companion Emergence Series. References [1] P. Cooney, Ordered Dynamics and Operational State Spaces, Paper I of the Ordered-Dynamics Reconstruction Program (2025). [2] P. Cooney, Records, Measurement, and Irreversibility, Paper IX of the Ordered-Dynamics Reconstruction Program (2025). [3] S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976). [4] J. D. Bekenstein, Universal upper bound on the entropy-to-energy ratio for bounded systems, Phys. Rev. D 23, 287 (1981). – 4 –